In this work, our main focus is on the so-called pairwise influence matrix, denoted as
\({{\mathcal {I}}}^{\Lambda ,\tau }_{G}\) . This is a central concept as the rapid mixing bounds we derive
using SI rely on showing that the maximum eigenvalue
of this matrix is bounded. We propose novel techniques that allow us to derive
more accurate estimations on the maximum eigenvalue of this matrix than what we have been
getting from the previous approaches that are proposed in [1]}, [2]}.
In turn, we get new rapid mixing results for the Glauber dynamics on two spin Gibbs distribution
such as the Hard-core model and the Ising model. Interestingly, in our results the
parameters of the Gibbs distribution do not depend on the maximum degree \({\Delta }\)  of the
underlying graph \(G\) . They rather depend on the spectral radius of the adjacency
matrix of \(G\) , denoted as \({\bf A}_G\) .
